Cotes's spiral

In physics and in the mathematics of plane curves, Cotes' spiral is a spiral that is typically written in one of three forms

${\frac {1}{r}}=A\cos \left(k\theta +\epsilon \right)$

${\frac {1}{r}}=A\cosh \left(k\theta +\epsilon \right)$

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where r and θ are the radius and azimuthal angle in a polar coordinate system, respectively, and A, k and ε are arbitrary real number constants. These spirals are named after Roger Cotes.

The significance of Cotes' spirals for physics are in the field of classical mechanics. These spirals are the solutions for the motion of a particle moving under a inverse-cube central force, e.g.,

$F(r)={\frac {\mu }{r^{3}}}$

where μ is any real number constant. A central force is one that depends only on the distance r between the moving particle and a point fixed in space, the center. In this case, the constant k of the spiral can be determined from μ and the areal velocity of the particle h by the formula

$k^{2}=1-{\frac {\mu }{h^{2}}}$

when μ < h² (cosine form of the spiral) and

$k^{2}={\frac {\mu }{h^{2}}}-1$

when μ > h² (hyperbolic cosine form of the spiral). When μ = h² exactly, the particle follows the third form of the spiral

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Bibliography

• Whittaker ET (1937). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies (4th ed. ed.). New York: Dover Publications. pp. pp. 80–83. ISBN 978-0-521-35883-5. {{cite book}}: |edition= has extra text (help); |pages= has extra text (help)

• Roger Cotes (1722) Harmonia Mensuarum, pp. 31, 98.

• Isaac Newton (1687) Philosophiæ Naturalis Principia Mathematica, Book I, §2, Proposition 9.

• Danby JM (1988). "The Case f(r) = μ/r³ — Cotes' Spiral (§4.7)". Fundamentals of Celestial Mechanics (2nd ed., rev. ed. ed.). Richmond, VA: Willmann-Bell. pp. pp. 69-71. ISBN 978-0943396200. {{cite book}}: |edition= has extra text (help); |pages= has extra text (help)

• Symon KR (1971). Mechanics (3rd ed. ed.). Reading, MA: Addison-Wesley. pp. p. 154. ISBN 978-0201073928. {{cite book}}: |edition= has extra text (help); |pages= has extra text (help)